A1
 
Dimensionless variables
for the channel electrode
 

In this appendix it is shown how the material balance equation may be reduced into a dimensionless form. A full treatment leads to the result that the dimensionless current may be expressed as a sole function of two mass transport parameters, a dimensionless time parameter and a number of dimensionless rate constants. If the Lévêque approximation is used to linearise the convective velocity profile, the two mass transport parameters reduce to a single one - the shear rate Peclet number.

A1.1 Full Treatment

A1.1.1 Steady-state transport-limited current

For a transport-limited electrolysis (for example a one electron reduction):

A + e => B(A1.1)

at a channel electrode the steady-state convective-diffusion equation is:

= 0 (A1.2)

If the dimensionless variables are introduced:

, (A1.3)

the convective-diffusion equation becomes:

(A1.4)

Under laminar flow conditions, when a Poiseille flow profile has fully developed, vx is given by:

(A1.5)
equation (A1.2) may be reduced into a dimensionless form:
(A1.6)

where:

and (A1.7)

Hence the concentration is a function of four parameters:

a = a(c,y,p1,p2)(A1.8)

The dimensionless current (Nusselt number) is given by:

(A1.9)

hence:

(A1.10)

c is set to zero and y disappears once the integral is evaluated, therefore the Nusselt number is purely a function of p1 and p2.

A1.1.2 Homogeneous kinetics

For an irreversible transport-limited ECE or EC2E reaction (written as a reduction):

A + e- = B
nB => C
C + e- = Products 		(A1.11)

(where n=1 defines an ECE reaction and n=2 defines an EC2E reaction), the steady-state material balance equations are:

= 0 (A1.12)
= 0 (A1.13)
= 0 (A1.14)

where kc is given by:

(A1.15)

These can be reduced into dimensionless forms in the same way. Species A does not have any kinetic terms in its material balance equation. The equation for species B becomes:

(A1.16)

The kinetic term is known as the dimensionless rate constant:

(A1.17)

Hence

b = b(c,y,p1,p2,Knorm)(A1.18)

Similarly the equation for species C becomes:

(A1.19)

Hence

c = c(c,y,p1,p2,Knorm,b) = c(c,y,p1,p2,Knorm)(A1.20)

Only species A and C are electroactive, so the dimensionless current is given by:

(A1.21)

The first integral has been given above and if the second is treated analogously:

(A1.22)

Where f and f' are used to represent different functions. Hence the ratio of the ECE current to the electron transfer current:

(A1.23)

A1.1.3 Time-dependent behaviour

If the dimensionless lengths are substituted into the time-dependent mass transport equation:
(A1.24)
thus:
(A1.25)
A dimensionless time, t, may be defined to absorb the leading term:
(A1.26)
where t is given by:
(A1.27)

A1.2 Lévêque Approximation

A1.2.1 Steady-state transport-limited current

If the Lévêque approximation is made:

(A1.28)

the dimensionless mass transport equation reduces to:

(A1.29)
where Ps is the shear rate Peclet number:
(A1.30)

thus:

a = a(c,y,Ps)(A1.31)

and Nu simplifies to a unique function of the shear Peclet number.

A1.2.2 Treatment without axial diffusion

If we redefine the dimensionless variable in the y co-ordinate:

(A1.32)

the mass transport equation becomes:

(A1.33)

which simplifies to:

(A1.34)

where:

p = Ps-2/3(A1.35)

If axial diffusion can be ignored, the equation reduces to:

(A1.36)

and the dependence on p is lost. The Nusselt number is a constant (from the integral) multiplied by Ps1/3.

(A1.37)

The constant may be found by solving equation (A1.37) analytically, giving the classical Levich equation1 :

(A1.38)

References

1 V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Engelwood Cliffs, New Jersey, (1962), p112.